中科院数学与系统科学研究院

数学研究所

学术报告

偏微分方程研讨班

报告人：Cristiana De Filippis (University of Parma)

题  目：Nonlinear potentials at the fractional scale:sharp regularity

时  间：2023.11.27（星期一）16:00-17:00

地  点： Meeting ID: 924 888 5804, Passcode: AMSS2022

摘 要：Nonlinear potential theory and elliptic regularity theory are two classical topics in the modern analysis of partial differential equations. In this talk I show how these themes merge to solve the longstanding open problem, dating back to the seminal contributions of O. A. Ladyzhenskaya & N. N. Ural’tseva, N. S. Trudinger, and L. Simon (1967-1976), of deriving Schauder estimates for minima of functionals (resp. solutions to elliptic equations) featuring polynomial nonuniform ellipticity. The sharp rate of nonuniform ellipticity for the validity of Schauder theory is also disclosed. From recent joint work with Giuseppe Mingione (Parma).

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报告人：Thomas Alazard (Ecole Normale Supérieure Paris-Saclay)

题  目：The Hele-Shaw semi-flow

时  间：2023.11.30（星期四）16:00-17:00

地  点：Meeting ID: 924 888 5804, Passcode: AMSS2022

摘 要：We give an elliptic formulation of the Hele-Shaw equation and use it to establish the well-posedness of the Cauchy problem in a comprehensive context. Notably, we prove that the Cauchy problem is well-posed in a strong sense and in a general setting, i.e. for initial data with barely a modulus of continuity, in any dimension. We will also study numerous qualitative properties, encompassing global regularity for initial data in sub-critical Sobolev spaces as well as waiting-time phenomena for Lipschitz solutions. This is a joint work with Herbert Koch.